cp's OEIS Frontend

A066522 Numbers n whose divisors less than or equal to sqrt(n) are consecutive, from 1 up to some number k.

%I A066522 #24 Feb 12 2017 17:39:13
%S A066522 1,2,3,4,5,6,7,8,10,11,12,13,14,17,18,19,22,23,24,26,29,31,34,37,38,
%T A066522 41,43,46,47,53,58,59,60,61,62,67,71,73,74,79,82,83,86,89,94,97,101,
%U A066522 103,106,107,109,113,118,122,127,131,134,137,139,142,146,149,151,157
%N A066522 Numbers n whose divisors less than or equal to sqrt(n) are consecutive, from 1 up to some number k.
%C A066522 The sequence consists of all numbers of the form p or 2p with p prime, along with 1, 8, 12, 18, 24 and 60. Sketch of proof: If k<=2 then n=1 or 8 or p or 2p. If k>2, then one of the numbers k+1, ..., k+4 is == 2 (mod 4); call it m. Then m/2 is an odd number <= k, so m = 2 * (m/2) divides n. Since m is not among 1,2,...,k, it must be greater than sqrt(n), so sqrt(n) < m <= k+4. Also, n is divisible by all positive integers <= k, including k, k-1 and k-2, whose least common multiple is their product divided by 1 or 2. So n >= k(k-1)(k-2)/2. Combining these inequalities implies k<=7 and n<=120.
%C A066522 Changing the definition to use "less than sqrt(n)" doesn't change the sequence. - _Stewart Gordon_, Sep 27 2011
%H A066522 Harry J. Smith, <a href="/A066522/b066522.txt">Table of n, a(n) for n = 1..1000</a>
%H A066522 J. G. van der Galien, <a href="http://www.home.zonnet.nl/galien8/index.html">The Dawn of Science</a>.
%e A066522 60 = 1*60 = 2*30 = 3*20 = 4*15 = 5*12 = 6*10.
%t A066522 test[n_] := Module[{}, d=Divisors[n]; d=Take[d, Ceiling[Length[d]/2]]; Last[d]==Length[d]]; Select[Range[1, 200], test]
%t A066522 cdQ[n_]:=Module[{d=Union[Differences[Select[Divisors[n],#<=Sqrt[n]&]]]},d=={}||d=={1}]; Select[Range[200],cdQ] (* _Harvey P. Dale_, Feb 12 2017 *)
%o A066522 (PARI) { n=0; for (m=1, 10^10, d=divisors(m); b=1; for (i=2, ceil(length(d)/2), if (d[i] - d[i-1] > 1, b=0; break)); if (b, write("b066522.txt", n++, " ", m); if (n==1000, return)) ) } \\ _Harry J. Smith_, Feb 21 2010
%o A066522 (Haskell)
%o A066522 import Data.List (genericLength)
%o A066522 a066522 n = a066522_list !! (n-1)
%o A066522 a066522_list = filter f [1..] where
%o A066522    f x = genericLength ds == maximum ds where ds = a161906_row x
%o A066522 -- _Reinhard Zumkeller_, Jun 24 2015, Nov 14 2011
%Y A066522 Cf. A066664 (composite terms); A074964, A000196.
%Y A066522 Cf. A161906.
%K A066522 nonn,nice,easy
%O A066522 1,2
%A A066522 Johan G. van der Galien (galien8(AT)zonnet.nl), Jan 05 2002
%E A066522 Edited by _Dean Hickerson_, Jan 07 2002